Pre-Calculus 30. PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions...
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- Pre-Calculus 30
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- PC30.10 PC30.10 Demonstrate understanding of polynomials and
polynomial functions of degree greater than 2 (limited to
polynomials of degree 5 with integral coefficients).
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- PC30.10 PC30.10 Demonstrate understanding of polynomials and
polynomial functions of degree greater than 2 (limited to
polynomials of degree 5 with integral coefficients).
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- What is a Polynomial Function?
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- A polynomial function has the form Where n = is a whole number
x = is a variable The coefficients a n to a 0 are real numbers The
degree of the poly function is n, the exponent of the greatest
power of x The leading coefficient is a n the coefficient of the
greatest power of x The constant term is a 0
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- Types of Polynomial Functions:
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- Each graph has at least one less change of direction then the
degree of its function.
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- Characteristics of Polynomial Functions:
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- Ex. 3.1 (p.114) #1-4 odds in each, 5-10 # 1-4 odds in each,
5-13 odds
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- PC30.10 PC30.10 Demonstrate understanding of polynomials and
polynomial functions of degree greater than 2 (limited to
polynomials of degree 5 with integral coefficients).
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- When only given the equation of a function what are some
strategies that we have use to find x-intercepts? Sub in a zero and
solve Factor Quadratic Formula Decompostion
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- Which of these strategies will work if our polynomial function
have a degree greater than 2?
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- In this section we will look at some methods to completely
factor a polynomial function with a degree greater than 2 in order
to find the zeros (x-intercepts).
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- Long Division Long division of a polynomial is done just like
your do with numbers but now you have variables
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- You use long division to divide a polynomial by a binomial: The
dividend, P(x), which is the polynomial that is being divided The
divisor, x-a, which is the binomial being divided into the
polynomial The quotient, Q(x), which is the expression that results
from the division The remainder, R, which is what is left over when
the division is done.
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- If we graphed the polynomial function from example 2 what would
the x-intercepts be?
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- Long division gives is factors of the polynomial function which
when set equal to zero and solved are x-intercepts, zeros, or
roots.
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- Synthetic Division: A short form of division that uses only the
coefficients of the terms It involves fewer calculations
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- Remainder Theorem: When a polynomial P(x) is divided by a
binomial x-a, the remainder is P(a) If the remainder is 0 then the
binomial x-a is a factor of P(x) If the remainder is not 0 then the
binomial x-a is NOT a factor of P(x)
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- Ex. 3.2 (p.124) #1,2,3-7 odds in each, 8-13 #2, 3-7 odds in
each, 8, 9-17 odds
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- PC30.10 PC30.10 Demonstrate understanding of polynomials and
polynomial functions of degree greater than 2 (limited to
polynomials of degree 5 with integral coefficients).
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- Last day we looked at how dividing polynomial functions by a
binomial shows us if that binomial is a factor or not We also
discussed how we want our equations in factored form because that
gives is the zeros/roots/x-intercepts Today we are going to extend
that idea
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- Factor Theorem: The factor theorem states that x-a is a factor
of a polynomial P(x) if and only if P(a)=0 If and only if means
that the result works both ways. That is, If x-a is a factor then
P(a)=0 If P(a)=0, then x-a is a factor of a polynomial P(x)
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- When factoring a polynomial function sometimes the most
difficult part is deciding which values of a we should use when
using long division, synthetic division or factor theorem.
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- This is referred to as the Integral Zero Theorem The integral
zero theorem describes the relationship between the factors and the
constant term of a polynomial. The theorem states that if x-a is a
factor of a polynomial P(x) with integral coefficients, then a is a
factor of the constant term of P(x) and x=a is a integral zero of
P(x).
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- Factor by Grouping: If a polynomial P(x) has an even number of
terms, it may be possible to group tow terms at a time and remove a
common factor If the binomial that results from common factoring is
the same for each pair of terms, then P(x) may be factored by
grouping Will not always work!!!!
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- Steps to factoring Polynomial Functions: 1. Use the integral
zero theorem to list possible integer values for zeros 2. You can
use the factor theorem to determine if the values that are zeros
(this take a lot of time so I dont suggest it) 2. Use one type of
division to determine all the factors 3. Write equation in factored
form
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- Ex. 3.3 (p.133) #1-7 odds in each, 8-14 evens #3-7 odds in
each, 8-16 evens
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- PC30.10 PC30.10 Demonstrate understanding of polynomials and
polynomial functions of degree greater than 2 (limited to
polynomials of degree 5 with integral coefficients).
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- We are now going to use all the information we have learned in
this unit to this point along with a little new into to Graph
Polynomial Functions without a graphing calculator
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- Ex. 3.4 (p.147) #1-10 odds in questions with multiple parts,
12-18 evens #3-10 odds in questions with multiple parts, 11-23
odds